# $d$-Galvin families

## TL;DR

This paper generalizes the Galvin problem by constructing small families of sets that partition the universe and are balanced on any subset, with implications for complexity theory.

## Contribution

It introduces a new generalization of the Galvin problem and provides polynomial-sized constructions for these families.

## Key findings

- Constructed polynomial-sized families for the generalized problem
- Families can partition the universe and are balanced on any subset
- Potential applications in complexity theory

## Abstract

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S \cap A| = |S \cap \overline{A}|$. We consider a generalization of this question that comes from a possible approach in complexity theory. In the generalization the required property is, for any $A$, to be able to find $d$ sets from a family $\mathcal{F} \subseteq \binom{[n]}{n/d}$ that form a partition of $[n]$ and such that each part is balanced on $A$. We construct such families of size polynomial in the parameters $n$ and $d$.